Compatibility conditions of continua using Riemann-Cartan geometry Christian G. B¨ohmer1 and Yongjo Lee2 September 9, 2020 Abstract The compatibility conditions for generalised continua are studied in the framework of differ- ential geometry, in particular Riemann-Cartan geometry. We show that Vall´ee’s compatibility condition in linear elasticity theory is equivalent to the vanishing of the three dimensional Ein- stein tensor. Moreover, we show that the compatibility condition satisfied by Nye’s tensor also arises from the three dimensional Einstein tensor which appears to play a pivotal role in contin- uum mechanics not mentioned before. We discuss further compatibility conditions which can be obtained using our geometrical approach and apply it to the micro-continuum theories. Keywords: compatibility conditions, Cosserat continuum, Riemann-Cartan geometry AMS 2010 subject classification: 74A35, 74A30, 53B50 1 Introduction Compatibility conditions in continuum mechanics form a set of partial differential equations which are not completely independent of each other. They may impose certain conditions among the unknown functions which are often derived by applying higher-order mixed partial derivatives to the given system of equations. They are closely related to integrability conditions. In 1992 Vall´ee [1] showed that the standard Saint-Venant compatibility condition of linear elasticity, known since the mid-19th century, can be written in the convenient form CurlΛ+CofΛ = 0 , (1.1) arXiv:2006.06800v2 [math-ph] 8 Sep 2020 where Λ is the 3 × 3 matrix given by 1 1 Λ= U(Curl U)T U − tr (Curl U)T U U . (1.2) det U 2 This formulation was based on Riemannian geometry where the metric tensor was written as gµν = a b Uµ Uν δab. Here U is the right stretch tensor of the polar decomposition of the deformation gradient tensor F = RU and R is an orthogonal matrix which is the polar part. The quantities Curl and Cof in (1.1) are defined by 1 (Curl U) = ǫ ∂ U , and (Cof U) = ǫ ǫ U U , (1.3) ij jmn m in ij 2 ims jnt mn st 1Christian G. B¨ohmer, Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK, email: [email protected] 2Yongjo Lee, Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK, email: [email protected] 1 and ǫijk is the totally skew-symmetric Levi-Civita symbol. Condition (1.1) was derived by finding the integrability condition of the system for the right Cauchy- Green deformation tensor C which is defined by C = (∇Θ)T (∇Θ) . (1.4) The deformation of the continuum is expressed by a diffeomorphism Θ : M → R3 such that x = X + u with u being the displacement vector. Hence, the tensor C assumes the role of a metric tensor in the given smooth manifold M. Later [2], the existence of such an immersion Θ is proved that maps an open subset of R3 into R3 in which the metric tensor field defined by C resides, given by U in the polar decomposition ∇Θ = RU. Equation (1.1) was shown to be equivalent to the vanishing of the Riemann curvature tensor in this setting. Much earlier, in 1953 Nye [3] showed that there exists a curvature related rank-two tensor Γ of the form 1 T T T Γ= tr R Curl R ½ − (R Curl R) , (1.5) 2 satisfying the compatibility condition CurlΓ+CofΓ = 0 . (1.6) The object Γ is often called Nye’s tensor and is written in terms of dislocation density tensor K = RT Curl R which only depends on the orthogonal matrix R. In this paper, we would like to show that these two compatibility conditions, seemingly arising from different and incomparable settings, are in fact special cases of a much broader compatibility condition which can be formulated in Riemann-Cartan geometry. Riemann-Cartan geometry provides a suitable background when one brings the concepts of cur- vature and torsion to the given manifold, using the method of differential geometry in describing the intrinsic nature of defects and its classifications. Pioneering works using this mathematical framework were explored in [4, 5, 6, 7, 8]. And many attempts to understand the theory of defects within the framework of the Einstein-Cartan theory were made in [9, 10, 11, 12, 13]. Curvature and torsion can be regarded as the sources for disclination and dislocation densities in the theory of defects respectively. The rotational symmetries are broken by the disclination and the translational symmetries are broken by the dislocation [7, 14, 15, 16] in Bravais lattices, the approximation of crystals into a continuum. It is worth noting that these geometries are commonly used in Einstein-Cartan theory [17, 18, 19], teleparallel gravity [20], gauge theories of gravity [21, 22, 23] and condensed matter system [24, 25, 26]. Links between micro-rotations and torsion were explored in [11, 27, 28, 29, 30]. Recent developments in incorporating elasticity theory and spin particles using the tetrad formalism can be found in [31, 32]. Our paper is organised as follows: In Section 2, after introducing frame bases and co-frame bases (also will be called as tangent and co-tangent bases) together with its polar decompositions, we define various quantities including a general connection, spin connection and torsion. We will see that the Riemann tensor can be expressed in various ways using above mentioned tensors. We introduce the Einstein tensor. Then we will decompose those tensors into two parts one which is torsion-free and one that contains torsion. In Section 3, using the tools introduced, we will derive compatibility conditions in various physical settings using a universal process. Firstly, Vall´ee’s result is re-derived, followed by Nye’s condition. We carefully explain the connection between these two compatibility conditions and the vanishing of the Einstein tensor. Furthermore, we will show that the Nye’s result is also closely linked to Skyrme theory and thus to micro-continuum theories. We briefly remark on the homotopic classification of the compatibility conditions. Section 4 derives general compatibility conditions based on our geometric approach followed by conclusions and discussions in the final section. 2 2 Tools of differential geometry 2.1 Frame fields Let us begin with a three-dimensional Riemannian manifold M with coordinates x and let us introduce a set of basis co-vectors (or 1-forms) for the co-tangent space at some point x ∈M 1 2 3 a {eµ,eµ,eµ} =: eµ(x) , (2.1) where the Latin indices a, b, . are tangent space indices and Greek letters µ, ν, . denote coordinate indices. This basis is often called a (co-)tetrad. The frame field consists of three orthogonal vector fields given by µ µ µ µ {E1 , E2 , E3 } =: Eb (x) . (2.2) These are dual basis satisfying the following orthogonality relations a ν ν a ν a eµEa = δµ , and eν Eb = δb . (2.3) ν a Here δµ and δb are the Kronecker deltas in their respective spaces. We emphasise that for a given manifold, we can find these tangent basis locally so that we can relate different sets of tangent bases in different points by simple transformations. However, it is impossible to find a single frame field which is nowhere vanishing globally, unless the manifold is parallelisable. For example, the hair ball problem illustrates that we cannot comb the hair on the 2-sphere S2 embedded in three dimensions smoothly everywhere. Hence the use of the locally defined diffeomorphism as the immersion of Θ : M → R3 used in (1.4). In the frame of tetrad formalism the metric tensor emerges as a secondary quantity defined in terms a of eµ. We have a b gµν = eµeν δab , (2.4) and recall that in Riemannian geometry this metric gives rise to an inner product between two vectors µ ν A · B := gµν A B , (2.5) which then naturally leads to a normed vector space. a This means we can use the co-tangent basis eµ to describe the deformation from the locally flat space δab given by the metric tensor gµν written in the coordinate basis. As a result, the metric a tensor gµν is obtained from the flat Euclidean metric δab by a set of deformations, governed by eµ at each point x ∈ M. Since any deformation can be regarded as a combination of rotation, shear and a compression, we can apply the polar decomposition to eµ as follows a a b eµ = R bUµ . (2.6) a b Here R b is an orthogonal matrix (a pure tangent space object) while the field Uµ is a symmetric and positive-definite matrix. Whenever we need to distinguish the micro-deformations from the macro- deformations, we will put a bar over the corresponding tensor. And in what follows we will often a b regard the matrix R b to be associated with micro-rotations, so that Uµ in the co-tangent basis can be T thought of as the first Cosserat deformation tensor [33]. This means U = R F . Hence the co-tangent basis is associated with the deformation gradient. When this decomposition is applied to (2.4) one arrives at a c d c d gµν = R cRadUµUν = δcdUµUν , (2.7) a b which shows that the metric is independent of R b and only depends on Uµ. This is a well-known result in differential geometry, namely the metric is independent of tangent space rotations. The polar decomposition for the inverse frame is µ b µ Ea = Ra Ub , (2.8) 3 µ a so that Ub is the inverse of Uµ , both of which are symmetric.